Euler's Substitutions for the Integral of a Particular Function

symbolic calculation

graphics aspect

substitution 1

substitution 2

substitution 3

condition

substitution

to square

apart/cancel

calculate x

calculate dx

show integral

∫Rx,

a

2

x

+bx+c

x

Euler's substitutions transform an integral of the form

∫Rx,

a

2

x

+bx+c

dx

, where

R

is a rational function of two arguments, into an integral of a rational function in the variable

t

. Euler's second and third substitutions select a point on the curve

2

y

=a

2

x

+bx+c

according to a method dependent on the parameter values and make

t

the parameter in the parametrized family of lines through that point. Euler's first substitution, used in the case where the curve is a hyperbola, lets

t

be the

y

intercept of a line parallel to one of the asymptotes of the curve. This Demonstration shows these curves and lines.

In symbolic calculations, the Demonstration shows:

1. If

a>0

, the substitution can be

a

2

x

+bx+c

=t±

a

x

. We only consider the case

t-

a

x

.

2. If

a

2

x

+bx+c=a(x-λ)(x-μ)

, where

λ

and

μ

are real numbers, the substitution is

a(x-λ)(x-μ)

=t(x-λ)

.

3. If

c>0

, the substitution can be

a

2

x

+bx+c

=xt±

c

. We only consider the case

xt+

c

.

In all three cases, a linear equation for

x

in terms of

t

is obtained. So

x

,

dx

, and

a

2

x

+bx+c

are rational expressions in

t

.